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<span class="kw">let </span><font color="Maroon" title="c1">U0</font> be   <a href="unialg_2.html#V2" title="UNIALG_2:attr.2">with_const_op</a>  <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span><br/> for <font color="Olive" title="b3">U1</font>, <font color="Olive" title="b4">U2</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font>  st <font color="Olive" title="b1">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">U1</font> &amp; <font color="Olive" title="b2">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">U2</font> holds <br/>( <font color="Olive" title="b1">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">l2</font> iff <font color="Olive" title="b3">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Olive" title="b4">U2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">l1</font>, <font color="Maroon" title="c3">l2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">U1</font>, <font color="Olive" title="b2">U2</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font>  st <font color="Maroon" title="c2">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">U1</font> &amp; <font color="Maroon" title="c3">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">U2</font> holds <br/>( <font color="Maroon" title="c2">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">l2</font> iff <font color="Olive" title="b1">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Olive" title="b2">U2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">U1</font>, <font color="Maroon" title="c5">U2</font> be   <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">U1</font> &amp; <font color="Maroon" title="c3">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">U2</font> implies ( <font color="Maroon" title="c2">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">l2</font> iff <font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c5">U2</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:26"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
( <font color="Maroon" title="c2">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">U1</font> &amp; <font color="Maroon" title="c3">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">U2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">l2</font> iff <font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c5">U2</font> )</span><br/>

<span class="kw">thus </span><a NAME="E2:26"/>
( <font color="Maroon" title="c2">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">l2</font> implies <font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c5">U2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c5">U2</font> implies <font color="Maroon" title="c2">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">l2</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:26_1"/>
<font color="Maroon" title="c2">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">l2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c5">U2</font></span><br/>

<a NAME="E2:26_1"/><span class="kw">then </span>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="unialg_3.html#T17" target="_self" title="UNIALG_3:th.17">Th17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:26_1"/>
<font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c5">U2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T14" title="UNIALG_2:th.14">UNIALG_2:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:26"/>
( <font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c5">U2</font> implies <font color="Maroon" title="c2">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">l2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:26_2"/>
<font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c5">U2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">l2</font></span><br/>

<a NAME="E2:26_2"/><span class="kw">then </span>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">U2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:26_2"/>
<font color="Maroon" title="c2">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="unialg_3.html#T17" target="_self" title="UNIALG_3:th.17">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


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